# Dictionary Definition

logically adv

1 according to logical reasoning; "logically, you
should now do the same to him"

2 in a logical manner; "he acted logically under
the circumstances" [ant: illogically]

# User Contributed Dictionary

# Extensive Definition

portal Logic Logic is the
study of the principles of valid inference and demonstration.
The word derives from Greek
λογική (logike), fem. of λογικός (logikos), "possessed of reason,
intellectual, dialectical, argumentative", from λόγος logos, "word, thought, idea,
argument, account, reason, or principle".

As a formal
science, logic investigates and classifies the structure of
statements and
arguments, both through
the study of formal
systems of inference and through the
study of arguments in natural language. The field of logic ranges
from core topics such as the study of validity, fallacies and paradoxes, to specialized
analysis of reasoning using probability and to arguments
involving causality.
Logic is also commonly used today in argumentation
theory.

Traditionally, logic was considered a branch of
philosophy, a part of
the classical trivium
of grammar, logic, and
rhetoric. Since the
mid-nineteenth century formal
logic has been studied in the context of foundations
of mathematics, where it was often called symbolic
logic. In 1879 Frege published
Begriffsschrift
: A formula language or pure thought modelled on that of
arithemetic which inaugurated modern logic with the invention
of quantifier
notation. In 1903 Alfred
North Whitehead and Bertrand
Russell attempted to establish logic formally as the
cornerstone of mathematics with the publication of Principia
Mathematica. However, except for the elementary part, the
system of Principia is no longer much used, having been largely
superseded by set theory. At
the same time the developments in the field of Logic since Frege, Russell
and Wittgenstein
had a profound influence on both the practice of philosophy and the
ideas concerning the nature of philosophical problems especially in
the English speaking world (see Analytic
philosophy). As the study of formal logic expanded, research no
longer focused solely on foundational issues, and the study of
several resulting areas of mathematics came to be called mathematical
logic. The development of formal logic and its implementation
in computing machinery is fundamental to computer
science. Logic is now widely taught by university philosophy
departments, more often than not as a compulsory discipline for
their students, especially in the English speaking world.

## Nature of logic

Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic logic is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions.- Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are a good example of informal logic.
- Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuance of natural language.)
- Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is often divided into two branches, propositional logic and predicate logic.
- Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

"Formal logic" is often used as a synonym for
symbolic logic, where informal logic is then understood to mean any
logical investigation that does not involve symbolic abstraction;
it is this sense of 'formal' that is parallel to the received
usages coming from "formal
languages" or "formal
theory". In the broader sense, however, formal logic is old,
dating back more than two millennia, while symbolic logic is
comparatively new, only about a century old.

### Consistency, soundness, and completeness

Among the valuable properties that logical systems can have are:-
- Consistency, which means that none of the theorems of the system contradict one another.
- Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.
- Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.

Not all systems achieve all three virtues. The
work of Kurt
Gödel has shown that no useful system of arithmetic can be both
consistent and complete: see
Gödel's incompleteness theorems. The Chinese philosopher
Gongsun
Long (ca. 325–250 BC) proposed
the paradox "One and one cannot become two, since neither becomes
two." In China, the tradition of scholarly investigation into
logic, however, was repressed by the Qin dynasty
following the legalist philosophy of Han
Feizi.

The first sustained work on the subject of logic
which has survived was that of Aristotle. The
formally sophisticated treatment of modern logic descends from the
Greek tradition, the latter mainly being informed from the
transmission of Aristotelian
logic.

Logic in Islamic philosophy also contributed to the development
of modern logic, which included the development of "Avicennian
logic" as an alternative to Aristotelian logic. Avicenna's system
of logic was responsible for the introduction of hypothetical
syllogism, temporal
modal
logic, and inductive
logic. The rise of the Asharite school,
however, limited original work on
logic in Islamic philosophy, though it did continue into the
15th century and had a significant influence on European logic
during the Renaissance.

In India, innovations in the scholastic school,
called Nyaya,
continued from ancient times into the early 18th
century, though it did not survive long into the colonial
period. In the 20th century, western philosophers like
Stanislaw Schayer and Klaus Glashoff have tried to explore certain
aspects of the Indian tradition
of logic. According to Hermann Weyl
(1929): Occidental mathematics has in past centuries broken away
from the Greek view and followed a course which seems to have
originated in India and which has been transmitted, with additions,
to us by the Arabs; in it the concept of number appears as
logically prior to the concepts of geometry.

During the medieval period, major efforts were
made to show that Aristotle's ideas were compatible with Christian faith.
During the later period of the Middle Ages, logic became a main
focus of philosophers, who would engage in critical logical
analyses of philosophical arguments.

## Topics in logic

### Syllogistic logic

The Organon was
Aristotle's
body of work on logic, with the Prior
Analytics constituting the first explicit work in formal logic,
introducing the syllogistic. The parts of syllogistic, also known
by the name term logic,
were the analysis of the judgements into propositions consisting of
two terms that are related by one of a fixed number of relations,
and the expression of inferences by means of syllogisms that consisted of
two propositions sharing a common term as premise, and a conclusion
which was a proposition involving the two unrelated terms from the
premises.

Aristotle's work was regarded in classical times
and from medieval times in Europe and the Middle East as the very
picture of a fully worked out system. It was not alone: the
Stoics
proposed a system of propositional
logic that was studied by medieval logicians; nor was the
perfection of Aristotle's system undisputed; for example the
problem of multiple generality was recognised in medieval
times. Nonetheless, problems with syllogistic logic were not seen
as being in need of revolutionary solutions.

Today, some academics claim that Aristotle's
system is generally seen as having little more than historical
value (though there is some current interest in extending term
logics), regarded as made obsolete by the advent of sentential
logic and the predicate
calculus. Others use Aristotle in argumentation
theory to help develop and critically question argumentation
schemes that are used in artificial
intelligence and legal
arguments.

### Predicate logic

Logic as it is studied today is a very different
subject to that studied before, and the principal difference is the
innovation of predicate logic. Whereas Aristotelian syllogistic
logic specified the forms that the relevant part of the involved
judgements took, predicate logic allows sentences to be analysed
into subject and argument in several different ways, thus allowing
predicate logic to solve the
problem of multiple generality that had perplexed medieval
logicians. With predicate logic, for the first time, logicians were
able to give an account of quantifiers general enough
to express all arguments occurring in natural language.

The development of predicate logic is usually
attributed to Gottlob
Frege, who is also credited as one of the founders of analytical
philosophy, but the formulation of predicate logic most often
used today is the first-order
logic presented in
Principles of Theoretical Logic by David
Hilbert and Wilhelm
Ackermann in 1928. The analytical
generality of the predicate logic allowed the formalisation of
mathematics, and drove the investigation of set theory,
allowed the development of Alfred
Tarski's approach to model
theory; it is no exaggeration to say that it is the foundation
of modern mathematical
logic.

Frege's original system of predicate logic was
not first-, but second-order. Second-order
logic is most prominently defended (against the criticism of
Willard
Van Orman Quine and others) by George
Boolos and Stewart
Shapiro.

### Modal logic

In languages, modality deals with the
phenomenon that sub-parts of a sentence may have their semantics
modified by special verbs or modal particles. For example, "We go
to the games" can be modified to give "We should go to the games",
and "We can go to the games"" and perhaps "We will go to the
games". More abstractly, we might say that modality affects the
circumstances in which we take an assertion to be satisfied.

The logical study of modality dates back to
Aristotle, who
was concerned with the alethic
modalities of necessity and possibility, which he observed to
be dual in the sense of De Morgan
duality. While the study of necessity and possibility remained
important to philosophers, little logical innovation happened until
the landmark investigations of Clarence
Irving Lewis in 1918, who formulated a
family of rival axiomatizations of the alethic modalities. His work
unleashed a torrent of new work on the topic, expanding the kinds
of modality treated to include deontic
logic and epistemic
logic. The seminal work of Arthur Prior
applied the same formal language to treat temporal
logic and paved the way for the marriage of the two subjects.
Saul
Kripke discovered (contemporaneously with rivals) his theory of
frame
semantics which revolutionised the formal technology available
to modal logicians and gave a new graph-theoretic
way of looking at modality that has driven many applications in
computational
linguistics and computer
science, such as dynamic
logic.

### Deduction and reasoning

The motivation for the study of logic in ancient
times was clear, as we have described: it is so that we may learn
to distinguish good from bad arguments, and so become more
effective in argument and oratory, and perhaps also, to become a
better person.

This motivation is still alive, although it no
longer takes centre stage in the picture of logic; typically
dialectical logic will
form the heart of a course in critical
thinking, a compulsory course at many universities, especially
those that follow the American model.

### Mathematical logic

Mathematical logic really refers to two distinct
areas of research: the first is the application of the techniques
of formal logic to mathematics and mathematical reasoning, and the
second, in the other direction, the application of mathematical
techniques to the representation and analysis of formal
logic.

The earliest use of mathematics and geometry in relation to logic
and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle. Many
other ancient and medieval philosophers applied mathematical ideas
and methods to their philosophical claims.

The boldest attempt to apply logic to mathematics
was undoubtedly the logicism pioneered by
philosopher-logicians such as Gottlob
Frege and Bertrand
Russell: the idea was that mathematical theories were logical
tautologies, and the programme was to show this by means to a
reduction of mathematics to logic. The various attempts to carry
this out met with a series of failures, from the crippling of
Frege's project in his Grundgesetze by Russell's
paradox, to the defeat of Hilbert's
program by
Gödel's incompleteness theorems.

Both the statement of Hilbert's program and its
refutation by Gödel depended upon their work establishing the
second area of mathematical logic, the application of mathematics
to logic in the form of proof
theory. Despite the negative nature of the incompleteness
theorems,
Gödel's completeness theorem, a result in model theory
and another application of mathematics to logic, can be understood
as showing how close logicism came to being true: every rigorously
defined mathematical theory can be exactly captured by a
first-order logical theory; Frege's proof
calculus is enough to describe the whole of mathematics, though
not equivalent to it. Thus we see how complementary the two areas
of mathematical logic have been.

If proof theory
and model theory
have been the foundation of mathematical logic, they have been but
two of the four pillars of the subject. Set theory
originated in the study of the infinite by Georg
Cantor, and it has been the source of many of the most
challenging and important issues in mathematical logic, from
Cantor's
theorem, through the status of the Axiom of
Choice and the question of the independence of the continuum
hypothesis, to the modern debate on large
cardinal axioms.

Recursion
theory captures the idea of computation in logical and arithmetic terms; its most
classical achievements are the undecidability of the Entscheidungsproblem
by Alan
Turing, and his presentation of the Church-Turing
thesis. Today recursion theory is mostly concerned with the
more refined problem of complexity
classes — when is a problem efficiently solvable? — and the
classification of degrees of
unsolvability.

### Philosophical logic

Philosophical
logic deals with formal descriptions of natural language. Most
philosophers assume that the bulk of "normal" proper reasoning can
be captured by logic, if one can find the right method for
translating ordinary language into that logic. Philosophical logic
is essentially a continuation of the traditional discipline that
was called "Logic" before the invention of mathematical logic.
Philosophical logic has a much greater concern with the connection
between natural language and logic. As a result, philosophical
logicians have contributed a great deal to the development of
non-standard logics (e.g., free logics,
tense
logics) as well as various extensions of classical
logic (e.g., modal
logics), and non-standard semantics for such logics (e.g.,
Kripke's
technique of supervaluations in the semantics of logic).

Logic and the philosophy of language are closely
related. Philosophy of language has to do with the study of how our
language engages and interacts with our thinking. Logic has an
immediate impact on other areas of study. Studying logic and the
relationship between logic and ordinary speech can help a person
better structure their own arguments and critique the arguments of
others. Many popular arguments are filled with errors because so
many people are untrained in logic and unaware of how to correctly
formulate an argument.

### Logic and computation

Logic cut to the heart of computer science as it
emerged as a discipline: Alan Turing's
work on the Entscheidungsproblem
followed from Kurt
Gödel's work on the incompleteness
theorems, and the notion of general purpose computers that came
from this work was of fundamental importance to the designers of
the computer machinery in the 1940s.

In the 1950s and 1960s, researchers predicted
that when human knowledge could be expressed using logic with
mathematical
notation, it would be possible to create a machine that
reasons, or artificial intelligence. This turned out to be more
difficult than expected because of the complexity of human
reasoning. In logic
programming, a program consists of a set of axioms and rules.
Logic programming systems such as Prolog compute the
consequences of the axioms and rules in order to answer a
query.

Today, logic is extensively applied in the fields
of artificial
intelligence, and computer
science, and these fields provide a rich source of problems in
formal and informal logic. Argumentation
theory is one good example of how logic is being applied to
artificial intelligence. The
ACM Computing Classification System in particular regards:

- Section F.3 on Logics and meanings of programs and F. 4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formal semantics of programming languages, as well as work of formal methods such as Hoare logic
- Boolean logic as fundamental to computer hardware: particularly, the system's section B.2 on Arithmetic and logic structures;
- Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic and default logic in Knowledge representation formalisms and methods, Horn clauses in logic programming, and description logic.

Furthermore, computers can be used as tools for
logicians. For example, in symbolic logic and mathematical logic,
proofs by humans can be computer-assisted. Using automated
theorem proving the machines can find and check proofs, as well
as work with proofs too lengthy to be written out by hand.

### Argumentation theory

Argumentation
theory is the study and research of informal logic, fallacies,
and critical questions as they relate to every day and practical
situations. Specific types of dialogue can be analyzed and
questioned to reveal premises, conclusions, and fallacies.
Argumentation theory is now applied in artificial
intelligence and law.

## Controversies in logic

Just as we have seen there is disagreement over
what logic is about, so there is disagreement about what logical
truths there are.

### Bivalence and the law of the excluded middle

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into the true and the false propositions. Systems which reject bivalence are known as non-classical logics.In 1910 Nicolai
A. Vasiliev rejected the law of excluded middle and the law of
contradiction and proposed the law of excluded fourth and logic
tolerant to contradiction. In the early 20th century
Jan
Łukasiewicz investigated the extension of the traditional
true/false values to include a third value, "possible", so
inventing ternary
logic, the first multi-valued
logic.

Logics such as fuzzy logic
have since been devised with an infinite number of "degrees of
truth", represented by a real number
between 0 and 1.

Intuitionistic
logic was proposed by L.E.J.
Brouwer as the correct logic for reasoning about mathematics,
based upon his rejection of the
law of the excluded middle as part of his intuitionism. Brouwer
rejected formalisation in mathematics, but his student Arend
Heyting studied intuitionistic logic formally, as did Gerhard
Gentzen. Intuitionistic logic has come to be of great interest
to computer scientists, as it is a constructive
logic, and is hence a logic of what computers can do.

Modal logic
is not truth conditional, and so it has often been proposed as a
non-classical logic. However, modal logic is normally formalised
with the principle of the excluded middle, and its relational
semantics is bivalent, so this inclusion is disputable. On the
other hand, modal logic can be used to encode non-classical logics,
such as intuitionistic logic.

Bayesian
probability can be interpreted as a system of logic where
probability is the subjective truth value.

### Implication: strict or material?

It is obvious that the notion of implication
formalised in classical logic does not comfortably translate into
natural language by means of "if… then…", due to a number of
problems called the paradoxes of material implication.

The first class of paradoxes involves
counterfactuals, such as "If the moon is made of green cheese, then
2+2=5", which are puzzling because natural language does not
support the principle
of explosion. Eliminating this class of paradoxes was the
reason for C. I.
Lewis's formulation of strict
implication, which eventually led to more radically revisionist
logics such as relevance
logic.

The second class of paradoxes involves redundant
premises, falsely suggesting that we know the succedent because of
the antecedent: thus "if that man gets elected, granny will die" is
materially true if granny happens to be in the last stages of a
terminal illness, regardless of the man's election prospects. Such
sentences violate the Gricean
maxim of relevance, and can be modelled by logics that reject
the principle of monotonicity
of entailment, such as relevance logic.

### Tolerating the impossible

Closely related to questions arising from the
paradoxes of implication comes the radical suggestion that logic
ought to tolerate inconsistency. Relevance
logic and paraconsistent
logic are the most important approaches here, though the
concerns are different: a key consequence of classical
logic and some of its rivals, such as intuitionistic
logic, is that they respect the principle
of explosion, which means that the logic collapses if it is
capable of deriving a contradiction. Graham
Priest, the main proponent of dialetheism, has argued for
paraconsistency on the grounds that there are in fact, true
contradictions.

### Is logic empirical?

What is the epistemological status of the laws of logic? What sort of argument is appropriate for criticising purported principles of logic? In an influential paper entitled "Is logic empirical?" Hilary Putnam, building on a suggestion of W.V. Quine, argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity, substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.Another paper by the same name by Sir
Michael Dummett argues that Putnam's desire for realism
mandates the law of distributivity. Distributivity of logic is
essential for the realist's understanding of how propositions are
true of the world in just the same way as he has argued the
principle of bivalence is. In this way, the question, "Is logic
empirical?" can be seen to lead naturally into the fundamental
controversy in metaphysics on
realism versus anti-realism.

## Notes

## References

- Brookshear, J. Glenn (1989), Theory of computation : formal languages, automata, and complexity, Benjamin/Cummings Pub. Co., Redwood City, Calif. ISBN 0805301437
- Cohen, R.S, and Wartofsky, M.W. (1974), Logical and Epistemological Studies in Contemporary Physics, Boston Studies in the Philosophy of Science, D. Reidel Publishing Company, Dordrecht, Netherlands. ISBN 90-277-0377-9.
- Finkelstein, D. (1969), "Matter, Space, and Logic", in R.S. Cohen and M.W. Wartofsky (eds. 1974).
- Gabbay, D.M., and Guenthner, F. (eds., 2001–2005), Handbook of Philosophical Logic, 13 vols., 2nd edition, Kluwer Publishers, Dordrecht.
- Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8.
- Hilbert, D., and Ackermann, W. (1928), Grundzüge der theoretischen Logik (Principles of Theoretical Logic), Springer-Verlag. OCLC 2085765
- Hodges, W. (2001), Logic. An introduction to Elementary Logic, Penguin Books.
- Hofweber, T. (2004), "Logic and Ontology", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
- Hughes, R.I.G. (ed., 1993), A Philosophical Companion to First-Order Logic, Hackett Publishing.
- Kneale, William, and Kneale, Martha, (1962), The Development of Logic, Oxford University Press, London, UK.
- Mendelson, Elliott (1964), Introduction to Mathematical Logic, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, Calif. OCLC 13580200
- Smith, B. (1989), "Logic and the Sachverhalt", The Monist 72(1), 52–69.
- Whitehead, Alfred North and Bertrand Russell (1910), Principia Mathematica, The University Press, Cambridge, England. OCLC 1041146

## Further reading

- The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject:
- Carroll,
Lewis
- "The Game of Logic", 1886. http://www.cut-the-knot.org/LewisCarroll/index.shtml
- [http://durendal.org:8080/lcsl/ "Symbolic Logic"], 1896.

- Samuel D. Guttenplan, Samuel D., Tamny, Martin, "Logic, a Comprehensive Introduction", Basic Books, 1971.
- Scriven, Michael, "Reasoning", McGraw-Hill, 1976, ISBN 0-07-055882-5
- Susan Haack. (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism, University of Chicago Press.
- Nicolas Rescher. (1964). Introduction to Logic, St. Martin's Press.

## See also

- Aristotle
- Artificial intelligence
- Deductive reasoning
- Digital electronics (also known as digital logic)
- Indian Logic
- Inductive reasoning
- Logic puzzle
- Logical consequence
- Mathematical logic
- Mathematics
- Philosophy
- Probabilistic logic
- Propositional logic
- Reason
- Straight and Crooked Thinking (book)
- Table of logic symbols
- Term logic
- Truth

## External links

- An Introduction to Philosophical Logic, by Paul Newall, aimed at beginners.
- arm-chair logic – the art of logical thinking Test your logic skills.
- forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic.
- Logic Self-Taught: A Workbook (originally prepared for on-line logic instruction).
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
- Translation Tips, by Peter Suber, for translating from English into logical notation.

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